Approximating gene transcription dynamics using steady-state formulas

Feng Jiao, Genghong Lin, and Jianshe Yu

Phys. Rev. E 104, 014401 – Published 2 July 2021

Abstract

Understanding how genes in a single cell respond to dynamically changing signals has been a central question in stochastic gene transcription research. Recent studies have generated massive steady-state or snapshot mRNA distribution data of individual cells, and inferred a large spectrum of kinetic transcription parameters under varying conditions. However, there have been few algorithms to convert these static data into the temporal variation of kinetic rates. Real-time imaging has been developed to monitor stochastic transcription processes at the single-cell level, but the immense technicality has prevented its application to most endogenous loci in mammalian cells. In this article, we introduced a stochastic gene transcription model with variable kinetic rates induced by unstable cellular conditions. We approximated the transcription dynamics using easily obtained steady-state formulas in the model. We tested the approximation against experimental data in both prokaryotic and eukaryotic cells and further solidified the conditions that guarantee the robustness of the method. The method can be easily implemented to provide convenient tools for quantifying dynamic kinetics and mechanisms underlying the widespread static transcription data, and may shed a light on circumventing the limitation of current bursting data on transcriptional real-time imaging.

Received 18 January 2021
Revised 11 May 2021
Accepted 2 June 2021

DOI:https://doi.org/10.1103/PhysRevE.104.014401

Physics Subject Headings (PhySH)

Gene expression

Stochastic dynamical systems

Master equation

Physics of Living SystemsNonlinear Dynamics

Authors & Affiliations

Feng Jiao ^1,2, Genghong Lin^1,2, and Jianshe Yu ^1,*

¹Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, People's Republic of China
²College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 51006, People's Republic of China

^*Corresponding author: jsyu@gzhu.edu.cn.

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Vol. 104, Iss. 1 — July 2021

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Images

Figure 1
Approximation to the transcription dynamics. Kinetic rates $v (t), δ (t), λ (t)$ , and $γ (t)$ were equipped with different dynamic profiles for the two-state model (5) with time-dependent rates. The approximation curves were generated by computing Eqs. (6) and (7), while the simulation curves referred to the numerical solutions of the two-state model (5) for $M (t), ϕ (t)$ , and $P_{m} (t)$ . When kinetic rates assumed (a) monotonic dynamics or (c) periodic dynamics, the approximation curves stayed very close to the corresponding simulation curves as in (b) or (d), respectively.
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Figure 2
Approximation to transcription dynamics in E. coli cells. (a) The degradation rate $δ$ of E. coli genes ranged from 0.25 to 1.1 $\min^{- 1}$ [2, 43]. The dynamic inactivation rate $γ (t)$ decreased exponentially and reached the steady state within 20–90 min. Under each pair of $δ$ and the duration of $γ (t)$ , the approximation curve and simulation curve for average mRNA levels and Fano factor were plotted, along with $R^{2}$ to quantify the approximation accuracy. (b) The agreement between the approximation and simulation curves ( $R^{2} > 0.9$ ) for transcription dynamics in E. coli cells. Other parameter values: $v = 15$ and $λ (t) = 0.27$ ( $\min^{- 1}$ ) [2].
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Figure 3
Approximation to transcription dynamics of Oct4 in mouse embryonic stem cells. (a) The dynamic activation rate $λ (t)$ changed dramatically in a short period after gene replication with other constant parameters $γ = 0.018 \min^{- 1}$ , $v = 1.9 \min^{- 1}$ for Oct4 [23, 45]. (b) The approximation curves (red dashed lines) and simulation curves (black solid lines) for the transcription dynamics (average level and Fano factor) were generated for each value of $δ$ , with their deviations being cross quantified using $R^{2}$ and MRE. (c) $R^{2}$ increased and MRE decreased dramatically with respect to $δ$ when $δ < 0.1 \min^{- 1}$ , and maintained $R^{2} > 0.9$ and $MRE < 10 %$ when $δ > 0.2 \min^{- 1}$ for the average level and $δ > 0.05 \min^{- 1}$ for the Fano factor.
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Figure 4
Recovering transcription dynamics via the steady-state data of E. coli. (a) The scattered steady-state data of Fano factor $ϕ$ vs mRNA mean $M$ was fitted by decreasing the inactivation rate $γ$ , from $γ = 400 \min^{- 1}$ under the lowest signal and $S_{0}$ , to $γ = 0.25 \min^{- 1}$ under the highest signal $S_{1}$ . (b) Under the real-time external change from $S_{0}$ to $S_{1}$ or from $S_{1}$ to $S_{0}$ , $γ (t)$ varied monotonically between 0.25 $\min^{- 1}$ and 400 $\min^{- 1}$ with different dynamic profiles. Accordingly, the embedded mRNA mean $M (t)$ and Fano factor $ϕ (t)$ assumed entirely different dynamics. (c) Each $γ (t)$ generated a curve of $ϕ (t)$ vs $M (t)$ . All the generated dynamic curves were clustered and the steady-state data were within a small neighborhood of the cluster. Green markers: Steady-state data from Lac promotor in E. coli cells under different concentrations of IPTG and cAMP [2]. Other parameter values: $δ = 0.5, v = 30.5, λ = 0.475$ ( $\min^{- 1}$ ) [2, 18].
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Figure 5
Slowing down the $γ (t)$ variation improves the correlation between the transcription dynamics and the steady-state data. (a) $γ (t)$ decreased exponentially, showing different durations to reach the steady state. (b) Each $γ (t)$ generated a dynamic curve of Fano factor $ϕ (t)$ vs mRNA mean $M (t)$ . Green circles represent steady-state data of $ϕ$ vs $M$ from the Lac promoter under different IPTG and cAMP concentrations [2]. The deviation between the dynamic curve and the fitting curve of the steady-state data was large when the duration of $γ (t)$ was relatively small. (c) The agreement rate was quantified using $R^{2}$ and MRE, which maintained $R^{2} > 0.9$ and $MRE < 10 %$ when the duration was longer than 20 min and 60 min, respectively.
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Figure 6
Predicting time-dependent rates $λ (t)$ and $γ (t)$ from the snapshot data. (a) Fitting discrete sets of snapshot data (black circles) established smooth dynamic curves for the average mRNA level $M (t)$ and Fano factor $ϕ (t)$ (black dashed lines). (b) Predicting $λ (t)$ and $γ (t)$ by substituting $M (t)$ and $ϕ (t)$ , with constant $v / δ$ and $δ$ , into the approximation formula (9). The prediction is confident when $δ = 0.5 \min^{- 1}$ and transcription duration $= 140$ min, as the transcription dynamics of the two-state model (5) with the predicted $λ (t)$ and $γ (t)$ (red lines) showed good correlation with the snapshot data in (a). (c) The regions (marked by red) of $δ$ and duration that guaranteed confident prediction ( $R^{2} > 0.9$ , $MRE < 10 %$ ). The other parameter: $v / δ = 60$ .
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